If he had to describe himself in five words, he'd say he's curious. And then forget to give you four more words.

There is an implicit "would" before "forget" in the second sentence, but it is usually dropped using ellipsis. This is why it's not "forgets". Putting the "would" in is still correct, but sounds overly formal to me.

To be strictly grammatically correct, you should say, "If he had to describe himself in 5 words, he would say he’s curious. He would then forget to give you 4 more words." Your original second sentence has no subject. But, especially in speech and informat writing, it's not uncommon to leave out the subject when it's the same as in the preceding sentence, especially if you want to indicate a rapid or abrupt sequence of events. "Bob heard the question. Answered it. Left the room."

You shouldn't say "he'd say ... and then forgets" because "he'd" is a contraction of "he would", so you're switching from future conditional, "would say", to basic present tense, "forgets". The technically correct construct would be to match the tenses of the verb, "would say ... would forget". You can get away with dropping the second "would" as you're alrady dropping the second subject, but you shouldn't change endings or add words to explicitly shift the tense, like saying "will forget" or "forgot" or "forgets".

Well, you could always shift the whole thing to another tense, for example past: "When he had to describe himself in 5 words, he said he was curious. He then forgot to give 4 more words." Etc.